Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 18.1s

Alternatives: 21

Speedup: 2.9×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1 + \beta}{t\_1}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\frac{t\_1}{\beta}}}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 5e+143)
     (* (+ alpha 1.0) (/ (/ (+ 1.0 beta) t_1) (* t_1 t_0)))
     (/ (/ (/ (+ alpha 1.0) (/ t_1 beta)) t_0) t_1))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+143) {
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	} else {
		tmp = (((alpha + 1.0) / (t_1 / beta)) / t_0) / t_1;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 5d+143) then
        tmp = (alpha + 1.0d0) * (((1.0d0 + beta) / t_1) / (t_1 * t_0))
    else
        tmp = (((alpha + 1.0d0) / (t_1 / beta)) / t_0) / t_1
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+143) {
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	} else {
		tmp = (((alpha + 1.0) / (t_1 / beta)) / t_0) / t_1;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+143:
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0))
	else:
		tmp = (((alpha + 1.0) / (t_1 / beta)) / t_0) / t_1
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+143)
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(Float64(1.0 + beta) / t_1) / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(t_1 / beta)) / t_0) / t_1);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+143)
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	else
		tmp = (((alpha + 1.0) / (t_1 / beta)) / t_0) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+143], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$1 / beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000012e143

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\alpha + 1\right) \cdot \color{blue}{\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)} \cdot \color{blue}{\left(\alpha + 1\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right), \color{blue}{\left(\alpha + 1\right)}\right) \]

   6. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\alpha + 1\right)} \]

   if 5.00000000000000012e143 < beta

   1. Initial program 73.5%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right), \color{blue}{\beta}\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 99.9%

      \[\leadsto \frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\color{blue}{\beta}}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\frac{t\_0}{\beta}}}{\alpha + \left(\beta + 3\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.8e+15)
     (/
      (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 2.0)))
      (+ 1.0 (+ 2.0 (+ alpha beta))))
     (/ (/ (/ (+ alpha 1.0) (/ t_0 beta)) (+ alpha (+ beta 3.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.8e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = (((alpha + 1.0) / (t_0 / beta)) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.8d+15) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 2.0d0))) / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = (((alpha + 1.0d0) / (t_0 / beta)) / (alpha + (beta + 3.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.8e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = (((alpha + 1.0) / (t_0 / beta)) / (alpha + (beta + 3.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.8e+15:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = (((alpha + 1.0) / (t_0 / beta)) / (alpha + (beta + 3.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.8e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(t_0 / beta)) / Float64(alpha + Float64(beta + 3.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.8e+15)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = (((alpha + 1.0) / (t_0 / beta)) / (alpha + (beta + 3.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.8e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$0 / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8e15

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 65.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

   5. Simplified 65.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 2.8e15 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right), \color{blue}{\beta}\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 3\right)\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 99.8%

      \[\leadsto \frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\color{blue}{\beta}}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \frac{\frac{\frac{\alpha + 1}{\frac{t\_0}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{t\_0} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (/ (/ (/ (+ alpha 1.0) (/ t_0 (+ 1.0 beta))) (+ alpha (+ beta 3.0))) t_0)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((alpha + 1.0) / (t_0 / (1.0 + beta))) / (alpha + (beta + 3.0))) / t_0;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = (((alpha + 1.0d0) / (t_0 / (1.0d0 + beta))) / (alpha + (beta + 3.0d0))) / t_0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((alpha + 1.0) / (t_0 / (1.0 + beta))) / (alpha + (beta + 3.0))) / t_0;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	return (((alpha + 1.0) / (t_0 / (1.0 + beta))) / (alpha + (beta + 3.0))) / t_0

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	return Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(t_0 / Float64(1.0 + beta))) / Float64(alpha + Float64(beta + 3.0))) / t_0)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = (((alpha + 1.0) / (t_0 / (1.0 + beta))) / (alpha + (beta + 3.0))) / t_0;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 95.6%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+r+ N/A

      \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-rgt1-in N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. distribute-lft-out N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ N/A

       \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]
7. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \frac{\frac{\alpha + 1}{t\_0}}{t\_0} \cdot \frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (/ (/ (+ alpha 1.0) t_0) t_0) (/ (+ 1.0 beta) (+ 3.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((alpha + 1.0) / t_0) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)));
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = (((alpha + 1.0d0) / t_0) / t_0) * ((1.0d0 + beta) / (3.0d0 + (alpha + beta)))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((alpha + 1.0) / t_0) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	return (((alpha + 1.0) / t_0) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	return Float64(Float64(Float64(Float64(alpha + 1.0) / t_0) / t_0) * Float64(Float64(1.0 + beta) / Float64(3.0 + Float64(alpha + beta))))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = (((alpha + 1.0) / t_0) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.6%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+r+ N/A

      \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-rgt1-in N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. distribute-lft-out N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ N/A

       \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]
7. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \beta\right)}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   4. associate-+r+ N/A

      \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   6. times-frac N/A

      \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 3\right)}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 2\right)}\right), \color{blue}{\left(\frac{1 + \beta}{\alpha + \left(\beta + 3\right)}\right)}\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{3 + \left(\alpha + \beta\right)}} \]
9. Add Preprocessing

Alternative 5: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 + \frac{\alpha + 2}{\beta}\right) \cdot \left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7e+15)
   (/
    (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 2.0)))
    (+ 1.0 (+ 2.0 (+ alpha beta))))
   (/
    (/ (* (+ -1.0 (/ (+ alpha 2.0) beta)) (- -1.0 alpha)) beta)
    (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7d+15) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 2.0d0))) / (1.0d0 + (2.0d0 + (alpha + beta)))
    else
        tmp = ((((-1.0d0) + ((alpha + 2.0d0) / beta)) * ((-1.0d0) - alpha)) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	} else {
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7e+15:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)))
	else:
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 + Float64(Float64(alpha + 2.0) / beta)) * Float64(-1.0 - alpha)) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7e+15)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + (2.0 + (alpha + beta)));
	else
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 + N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7e15

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \beta}{{\left(2 + \beta\right)}^{2}}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, 1\right)}\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, 1\right)\right), 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 65.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{1}\right)\right), 1\right)\right) \]

   5. Simplified 65.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 7e15 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 91.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 91.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2\right) + 1} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 1\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 3} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \color{blue}{\left(\beta + 3\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\color{blue}{\alpha} + \left(\beta + 3\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      13. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      16. +-lowering-+.f64 91.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right)\right)\right) \]

   7. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{3 + \left(\alpha + \beta\right)}} \]

   8. Taylor expanded in beta around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\beta}\right)}, \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\beta}\right)\right), \mathsf{+.f64}\left(\color{blue}{3}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\mathsf{neg}\left(\beta\right)}\right), \mathsf{+.f64}\left(\color{blue}{3}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{-1 \cdot \beta}\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(\color{blue}{3}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot -1 + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot -1 + \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(-1 + \frac{2 + \alpha}{\beta}\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(-1 + \frac{2 + \alpha}{\beta}\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(-1 + \frac{2 + \alpha}{\beta}\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \left(\frac{2 + \alpha}{\beta}\right)\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(2 + \alpha\right), \beta\right)\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \beta\right)\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \beta\right)\right)\right), \left(\mathsf{neg}\left(\beta\right)\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      14. neg-lowering-neg.f64 90.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \beta\right)\right)\right), \mathsf{neg.f64}\left(\beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   10. Simplified 90.8%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(-1 + \frac{2 + \alpha}{\beta}\right)}{-\beta}}}{3 + \left(\alpha + \beta\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 + \frac{\alpha + 2}{\beta}\right) \cdot \left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{t\_0} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 + \frac{\alpha + 2}{\beta}\right) \cdot \left(-1 - \alpha\right)}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ alpha beta))))
   (if (<= beta 1.75e+15)
     (* (/ (+ 1.0 beta) t_0) (/ 1.0 (* (+ beta 2.0) (+ beta 2.0))))
     (/ (/ (* (+ -1.0 (/ (+ alpha 2.0) beta)) (- -1.0 alpha)) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.75e+15) {
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (alpha + beta)
    if (beta <= 1.75d+15) then
        tmp = ((1.0d0 + beta) / t_0) * (1.0d0 / ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((((-1.0d0) + ((alpha + 2.0d0) / beta)) * ((-1.0d0) - alpha)) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 1.75e+15) {
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (alpha + beta)
	tmp = 0
	if beta <= 1.75e+15:
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1.75e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 + Float64(Float64(alpha + 2.0) / beta)) * Float64(-1.0 - alpha)) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 1.75e+15)
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = (((-1.0 + ((alpha + 2.0) / beta)) * (-1.0 - alpha)) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.75e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 + N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.75e15

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]
   7. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \beta\right)}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. times-frac N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 3\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 2\right)}\right), \color{blue}{\left(\frac{1 + \beta}{\alpha + \left(\beta + 3\right)}\right)}\right) \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{3 + \left(\alpha + \beta\right)}} \]

   9. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{{\left(2 + \beta\right)}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \beta\right)}, \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\beta}\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\beta}\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       7. +-lowering-+.f64 65.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

   11. Simplified 65.0%

       \[\leadsto \color{blue}{\frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}} \cdot \frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \]

   if 1.75e15 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 91.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 91.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2\right) + 1} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 1\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 3} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \color{blue}{\left(\beta + 3\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\color{blue}{\alpha} + \left(\beta + 3\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      13. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      16. +-lowering-+.f64 91.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right)\right)\right) \]

   7. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{3 + \left(\alpha + \beta\right)}} \]

   8. Taylor expanded in beta around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\beta}\right)}, \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\beta}\right)\right), \mathsf{+.f64}\left(\color{blue}{3}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\mathsf{neg}\left(\beta\right)}\right), \mathsf{+.f64}\left(\color{blue}{3}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{-1 \cdot \beta}\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(1 + \alpha\right) + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(\color{blue}{3}, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot -1 + \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot -1 + \left(1 + \alpha\right) \cdot \frac{2 + \alpha}{\beta}\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \alpha\right) \cdot \left(-1 + \frac{2 + \alpha}{\beta}\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \alpha\right), \left(-1 + \frac{2 + \alpha}{\beta}\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(-1 + \frac{2 + \alpha}{\beta}\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \left(\frac{2 + \alpha}{\beta}\right)\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(2 + \alpha\right), \beta\right)\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \beta\right)\right)\right), \left(-1 \cdot \beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \beta\right)\right)\right), \left(\mathsf{neg}\left(\beta\right)\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

      14. neg-lowering-neg.f64 90.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \alpha\right), \beta\right)\right)\right), \mathsf{neg.f64}\left(\beta\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right) \]

   10. Simplified 90.8%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \alpha\right) \cdot \left(-1 + \frac{2 + \alpha}{\beta}\right)}{-\beta}}}{3 + \left(\alpha + \beta\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 + \frac{\alpha + 2}{\beta}\right) \cdot \left(-1 - \alpha\right)}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{t\_0} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ alpha beta))))
   (if (<= beta 8.2e+15)
     (* (/ (+ 1.0 beta) t_0) (/ 1.0 (* (+ beta 2.0) (+ beta 2.0))))
     (/ (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 8.2e+15) {
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (alpha + beta)
    if (beta <= 8.2d+15) then
        tmp = ((1.0d0 + beta) / t_0) * (1.0d0 / ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double tmp;
	if (beta <= 8.2e+15) {
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (alpha + beta)
	tmp = 0
	if beta <= 8.2e+15:
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 8.2e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 8.2e+15)
		tmp = ((1.0 + beta) / t_0) * (1.0 / ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8.2e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.2e15

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]
   7. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \beta\right)}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. times-frac N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 3\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(\beta + 2\right)}\right), \color{blue}{\left(\frac{1 + \beta}{\alpha + \left(\beta + 3\right)}\right)}\right) \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{3 + \left(\alpha + \beta\right)}} \]

   9. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{{\left(2 + \beta\right)}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left({\left(2 + \beta\right)}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \beta\right)}, \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\beta}\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\beta}\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

       7. +-lowering-+.f64 65.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right)\right) \]

   11. Simplified 65.0%

       \[\leadsto \color{blue}{\frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}} \cdot \frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \]

   if 8.2e15 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 91.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 91.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2\right) + 1} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 1\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 3} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \color{blue}{\left(\beta + 3\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\color{blue}{\alpha} + \left(\beta + 3\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      13. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      16. +-lowering-+.f64 91.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right)\right)\right) \]

   7. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{3 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8e+15)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) (+ alpha (+ beta 2.0))) (+ 3.0 (+ alpha beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.8d+15) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / (alpha + (beta + 2.0d0))) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8e+15) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4.8e+15:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / (3.0 + (alpha + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(alpha + Float64(beta + 2.0))) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8e+15)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha + 1.0) / (alpha + (beta + 2.0))) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4.8e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.8e15

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{\beta + 2}}{\color{blue}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{\beta + 2}\right), \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(\beta + 2\right)\right), \left(\color{blue}{\left(\beta + 2\right)} \cdot \left(\beta + 3\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\beta + 2\right)\right), \left(\left(\color{blue}{\beta} + 2\right) \cdot \left(\beta + 3\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(\beta + \color{blue}{2}\right) \cdot \left(\beta + 3\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\color{blue}{\beta} + 3\right)\right)\right) \]

      9. +-lowering-+.f64 63.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   9. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}} \]

   if 4.8e15 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 91.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 91.3%

      \[\leadsto \frac{\frac{\color{blue}{1 + \alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2\right) + 1} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 1\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 3} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \color{blue}{\left(\beta + 3\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}\right), \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\color{blue}{\alpha} + \left(\beta + 3\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \left(\alpha + \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \left(\beta + 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\alpha + \left(\beta + 3\right)\right)\right) \]

      13. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \left(3 + \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \color{blue}{\left(\alpha + \beta\right)}\right)\right) \]

      16. +-lowering-+.f64 91.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right), \mathsf{+.f64}\left(3, \mathsf{+.f64}\left(\alpha, \color{blue}{\beta}\right)\right)\right) \]

   7. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{3 + \left(\alpha + \beta\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4e+16)
   (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))
   (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4d+16) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4e+16) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 4e+16:
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4e+16)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4e+16)
		tmp = ((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0));
	else
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 4e+16], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4e16

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1 + \beta}{\left(\beta + 2\right) \cdot \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{\beta + 2}}{\color{blue}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \beta}{\beta + 2}\right), \color{blue}{\left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \beta\right), \left(\beta + 2\right)\right), \left(\color{blue}{\left(\beta + 2\right)} \cdot \left(\beta + 3\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\beta + 2\right)\right), \left(\left(\color{blue}{\beta} + 2\right) \cdot \left(\beta + 3\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\left(\beta + \color{blue}{2}\right) \cdot \left(\beta + 3\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\left(\beta + 2\right), \color{blue}{\left(\beta + 3\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\color{blue}{\beta} + 3\right)\right)\right) \]

      9. +-lowering-+.f64 63.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   9. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}} \]

   if 4e16 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 91.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 91.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.15e+17)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+17) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.15d+17) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.15e+17) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.15e+17:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.15e+17)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.15e+17)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.15e+17], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15e17

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   if 1.15e17 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, 1\right)\right)}, 1\right)\right) \]

      2. +-lowering-+.f64 91.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(\alpha, \beta\right)}, \mathsf{*.f64}\left(2, 1\right)\right), 1\right)\right) \]

   5. Simplified 91.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.6e+16)
   (/ (+ 1.0 beta) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.6e+16) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.6d+16) then
        tmp = (1.0d0 + beta) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.6e+16) {
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 5.6e+16:
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.6e+16)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.6e+16)
		tmp = (1.0 + beta) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.6e+16], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.6e16

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   if 5.6e16 < beta

   1. Initial program 84.1%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 91.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 91.2%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\alpha + \left(\beta + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.85)
   (+
    0.08333333333333333
    (*
     beta
     (+
      (* beta (+ (* beta 0.024691358024691357) -0.011574074074074073))
      -0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.85) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.85d0) then
        tmp = 0.08333333333333333d0 + (beta * ((beta * ((beta * 0.024691358024691357d0) + (-0.011574074074074073d0))) + (-0.027777777777777776d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.85) {
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.85:
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.85)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(Float64(beta * Float64(Float64(beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.85)
		tmp = 0.08333333333333333 + (beta * ((beta * ((beta * 0.024691358024691357) + -0.011574074074074073)) + -0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.85], N[(0.08333333333333333 + N[(beta * N[(N[(beta * N[(N[(beta * 0.024691358024691357), $MachinePrecision] + -0.011574074074074073), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8500000000000001

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \left(\mathsf{neg}\left(\frac{5}{432}\right)\right)\right)\right), \frac{-1}{36}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \left(\frac{2}{81} \cdot \beta + \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{2}{81} \cdot \beta\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

      11. *-lowering-*.f64 63.2%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{2}{81}\right), \frac{-5}{432}\right)\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 63.2%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)} \]

   if 1.8500000000000001 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\alpha + \left(\beta + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(\beta \cdot 0.024691358024691357 + -0.011574074074074073\right) + -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.56)
   (+
    0.08333333333333333
    (* beta (+ -0.027777777777777776 (* beta -0.011574074074074073))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.56) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.56d0) then
        tmp = 0.08333333333333333d0 + (beta * ((-0.027777777777777776d0) + (beta * (-0.011574074074074073d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.56) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.56:
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.56)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(-0.027777777777777776 + Float64(beta * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.56)
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.56], N[(0.08333333333333333 + N[(beta * N[(-0.027777777777777776 + N[(beta * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5600000000000001

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{-5}{432} \cdot \beta\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

      7. *-lowering-*.f64 63.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 63.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 1.5600000000000001 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\alpha + \left(\beta + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.56:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.65)
   (+
    0.08333333333333333
    (* beta (+ -0.027777777777777776 (* beta -0.011574074074074073))))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.65d0) then
        tmp = 0.08333333333333333d0 + (beta * ((-0.027777777777777776d0) + (beta * (-0.011574074074074073d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.65) {
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 1.65:
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.65)
		tmp = Float64(0.08333333333333333 + Float64(beta * Float64(-0.027777777777777776 + Float64(beta * -0.011574074074074073))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.65)
		tmp = 0.08333333333333333 + (beta * (-0.027777777777777776 + (beta * -0.011574074074074073)));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 1.65], N[(0.08333333333333333 + N[(beta * N[(-0.027777777777777776 + N[(beta * -0.011574074074074073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6499999999999999

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \left(\frac{-5}{432} \cdot \beta + \frac{-1}{36}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\frac{-5}{432} \cdot \beta\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\left(\beta \cdot \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

      7. *-lowering-*.f64 63.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\beta, \frac{-5}{432}\right), \frac{-1}{36}\right)\right)\right) \]

   10. Simplified 63.1%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot \left(\beta \cdot -0.011574074074074073 + -0.027777777777777776\right)} \]

   if 1.6499999999999999 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 86.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\beta}}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.65:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot -0.011574074074074073\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 63.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{\beta}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \alpha}{\beta}\right), \color{blue}{\beta}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \beta\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right), \beta\right) \]

      5. +-lowering-+.f64 86.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right), \beta\right) \]

   9. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\beta}}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 94.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (+ alpha 1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 63.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 92.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta (+ beta 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * (beta + 2.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.6], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 63.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.60000000000000009 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\alpha + \left(\beta + 2\right)} \]

   10. Taylor expanded in alpha around 0

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(2 + \beta\right)}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \color{blue}{\left(2 + \beta\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \left(\beta + \color{blue}{2}\right)\right)\right) \]

       4. +-lowering-+.f64 84.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(\beta, \color{blue}{2}\right)\right)\right) \]

   12. Simplified 84.8%

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \left(\beta + 2\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 92.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 63.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \alpha\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      4. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{\beta \cdot \beta}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      3. *-lowering-*.f64 84.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   10. Simplified 84.8%

       \[\leadsto \color{blue}{\frac{1}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 47.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.95)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.95d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.95) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 2.95:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.95)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.95)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 2.95], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9500000000000002

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12} + \frac{-1}{36} \cdot \beta} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{-1}{36} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \left(\beta \cdot \color{blue}{\frac{-1}{36}}\right)\right) \]

      3. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{-1}{36}}\right)\right) \]

   10. Simplified 63.0%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.9500000000000002 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\alpha + \left(\beta + 2\right)} \]

   10. Taylor expanded in alpha around inf

       \[\leadsto \color{blue}{\frac{1}{\beta}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 7.6%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\beta}\right) \]

   12. Simplified 7.6%

       \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 46.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   1. Initial program 99.8%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

      11. +-lowering-+.f64 63.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{12}} \]
   9. Step-by-step derivation

   10. Simplified 62.7%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 12 < beta

   1. Initial program 85.2%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + 3\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \color{blue}{\beta}\right) + 3\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + \color{blue}{1}\right)\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) + \color{blue}{1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\color{blue}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\right), \color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]

   7. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 + \alpha}{\beta}\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \alpha\right), \beta\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

      2. +-lowering-+.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \alpha\right), \beta\right), \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\beta, 2\right)\right)\right) \]

   9. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \alpha}{\beta}}}{\alpha + \left(\beta + 2\right)} \]

   10. Taylor expanded in alpha around inf

       \[\leadsto \color{blue}{\frac{1}{\beta}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 7.6%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\beta}\right) \]

   12. Simplified 7.6%

       \[\leadsto \color{blue}{\frac{1}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 45.2% accurate, 35.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 95.6%

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
2. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{1 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2} \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{\frac{1 + \left(\beta + \left(\alpha + \beta \cdot \alpha\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   5. associate-+r+ N/A

      \[\leadsto \frac{\frac{\left(1 + \beta\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\alpha + \beta \cdot \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   8. distribute-rgt1-in N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot 1 + \left(\beta + 1\right) \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot 1}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. distribute-lft-out N/A

      \[\leadsto \frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   10. associate-*l/ N/A

       \[\leadsto \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)\right), \color{blue}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}\right) \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
4. Add Preprocessing

5. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left({\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \left(\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left({\left(2 + \beta\right)}^{2}\right), \color{blue}{\left(3 + \beta\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 + \beta\right), \left(2 + \beta\right)\right), \left(\color{blue}{3} + \beta\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\beta + 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(2 + \beta\right)\right), \left(3 + \beta\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \left(\beta + 2\right)\right), \left(3 + \beta\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(3 + \beta\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \left(\beta + \color{blue}{3}\right)\right)\right) \]

   11. +-lowering-+.f64 66.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 2\right), \mathsf{+.f64}\left(\beta, 2\right)\right), \mathsf{+.f64}\left(\beta, \color{blue}{3}\right)\right)\right) \]

7. Simplified 66.1%

   \[\leadsto \color{blue}{\frac{1 + \beta}{\left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right) \cdot \left(\beta + 3\right)}} \]

8. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1}{12}} \]
9. Step-by-step derivation

10. Simplified 45.8%

    \[\leadsto \color{blue}{0.08333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))